Maximum principles for the fractional p-Laplacian and symmetry of solutions

TL;DR

This paper develops maximum principles and symmetry results for solutions to nonlinear equations involving the fractional p-Laplacian, a nonlocal operator, using the method of moving planes.

Contribution

It introduces a maximum principle for anti-symmetric functions and key boundary estimates, enabling symmetry and monotonicity results for solutions.

Findings

Proved maximum principle for anti-symmetric functions.

Established radial symmetry and monotonicity of solutions.

Developed methods applicable to various nonlinear nonlocal problems.

Abstract

In this paper, we consider nonlinear equations involving the fractional p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$ We prove a {\em maximum principle for anti-symmetric functions} and obtain other key ingredients for carrying on the method of moving planes, such as {\em a key boundary estimate lemma}. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.